https://file.aiquickdraw.com/mathbot/file/1729540988644x2g479si.jpg

The system of equations provided in the image is:

\[
\begin{cases}
y = \frac{1}{2}x - 3 \\
x - 2y = 6
\end{cases}
\]

To solve this system, we can use the **substitution method**.

### Step 1: Substitute \( y = \frac{1}{2}x - 3 \) into the second equation.

The second equation is:
\[
x - 2y = 6
\]
Substitute \( y = \frac{1}{2}x - 3 \) into this:
\[
x - 2\left(\frac{1}{2}x - 3\right) = 6
\]
Simplify the expression inside the parentheses:
\[
x - \left(x - 6\right) = 6
\]
\[
x - x + 6 = 6
\]
This simplifies to:
\[
6 = 6
\]
### Step 2: Interpretation of the result

Since the equation reduces to \( 6 = 6 \), which is a **true statement** but does not involve any variables, this indicates that the system of equations has **infinitely many solutions**. This happens when both equations represent the same line.

Therefore, the system has **infinitely many solutions**, meaning the two lines coincide.

### Graphical Interpretation:
If we were to graph these two equations, they would lie on the same line, reinforcing that there are infinitely many points where the lines intersect.

---

Would you like me to go into more detail or clarify any step further? 

Here are 5 follow-up questions for you:
1. How can you identify if a system has infinitely many solutions or no solution just from its equations?
2. What does it mean graphically when two lines coincide?
3. Can you think of an example of a system of equations that has no solution?
4. How would you solve this system using the elimination method?
5. What are the implications of the result "6 = 6" in terms of systems of equations?

**Tip:** When solving systems of linear equations, always look for ways to simplify the system early on, such as substitution or elimination methods, which can make the process faster.

Solve the system of equations:
 y = (1/2)x - 3 and x - 2y = 6.

Explore how to solve the system of linear equations y = (1/2)x - 3 and x - 2y = 6 using the substitution method. This tutorial includes a detailed step-by-step solution and explains why this system has infinitely many solutions. Suitable for Grades 9-11.

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